The regularity theory for the parabolic double obstacle problem

Abstract

In this paper, we study the regularity of the free boundaries of the parabolic double obstacle problem for the heat operator and fully nonlinear operator. The result in this paper are generalizations of the theory for the elliptic problem in Lee et al. (Calc Var Partial Differ Equ 58(3):104, 2019) and Lee and Park (The regularity theory for the double obstacle problem for fully nonlinear operator, , 2018) to parabolic case and also the theory for the parabolic single obstacle problem in Caffarelli et al. (J Am Math Soc 17(4):827–869, 2004) to double obstacle case. New difficulties in the theory which are generated by the characteristic of parabolic PDEs and the existence of the upper obstacle are discussed in detail. Furthermore, the thickness assumptions to have the regularity of the free boundary are carefully considered.

Appendix A: Appendices

In this appendix, we discuss the classification of the global solutions to the double obstacle problem in $Q_1$ with $0 \in \partial \Omega _u \cap \partial \Omega _u^\psi$, Proposition A.3. The proposition is used in the proof of the regularity for the free boundary of $u \in P_r(M)$ ( e.g. Lemma 5.1) and $u \in {\tilde{P}}_r(M)$. Furthermore, by Proposition A.3 and the methods for $P_r(M)$ and ${\tilde{P}}_r(M)$, we could have the regularity of the free boundary of $u \in {\tilde{P}}^+_\infty (M)$, see Definition A.1.

Frist, we define the solution spaces of the problem.

Definition A.1

(Local solutions) The class of local solutions ${\tilde{P}}^+_r(M)$ $(0<r<\infty )$ consists of all solutions u of

$$\begin{aligned} {\tilde{H}}u:= -\partial _t u+F(D^2u, x) =f\chi _{\Omega _u \cap \Omega ^\psi _u } + {\tilde{H}} \psi \chi _{\Omega _u \cap \Lambda ^\psi _u}, \quad 0\le u\le \psi \quad \text{ in } Q_r, \end{aligned}$$

for open sets $\Omega _u$ and $\Omega ^\psi _u$ such that $\Omega _{u} \supset \{u>0\}$ and $\Omega ^{\psi }_{u} \supset \{u< \psi \}$, with

• (i) $\Vert \partial _t u \Vert _{L^\infty (Q_r)}+\Vert D^2 u \Vert _{L^\infty (Q_r)} \le M$,

• (ii) $0\in \partial \Omega _u \cap \partial \Omega ^\psi _u,$

where $f\in C^{0,1}_x\cap C^{0,1}_t(Q_r)$, $\psi \in C^{1,1}_x \cap C^{0,1}_t(Q_r)$, and $D\psi , \partial _t \psi \in C^{1,1}_x(\overline{\Omega _\psi )})\cap C^{0,1}_t(\overline{\Omega _\psi })$, for an open set $\Omega _\psi \supset \{\psi >0\}$.

Definition A.2

(Global solutions) The class of global solutions ${\tilde{P}}^+_\infty (M)$ consists of all solutions u of

$$\begin{aligned} {\tilde{H}}u = \chi _{\Omega _u \cap \Omega ^\psi _u} + a \chi _{\Omega _u \cap \Lambda ^\psi _u}, \quad 0\le u \le \psi \quad \text { in } {\mathbb {R}}^n \times {\mathbb {R}}, \end{aligned}$$

for open sets $\Omega _u$ and $\Omega ^\psi _u$ such that $\Omega _{u} \supset \{u>0\}$, $\Omega ^{\psi }_{u} \supset \{u< \psi \}$, and a constant $a>1$ which satisfying:

• (i) ${\tilde{H}} \psi =a \chi _{\Omega (\psi )}$ in ${\mathbb {R}}^n\times {\mathbb {R}}$,

• (ii) $\Vert \partial _t u \Vert _{L^\infty ({\mathbb {R}}^n \times {\mathbb {R}})}+\Vert D^2 u \Vert _{L^\infty ({\mathbb {R}}^n \times {\mathbb {R}})} \le M$,

• (iii) $0\in \partial \Omega _u \cap \partial \Omega ^\psi _u.$

As in the cases $P_r(M)$ and ${\tilde{P}}_r(M)$, we first study the classification of the global solutions to the single obstacle problem:

$$\begin{aligned} -\partial _t u+ F(D^2 u,x) =f\chi _{\Omega _u}, \qquad&u\ge 0 \quad \text{ in } Q_1, \end{aligned}$$

(44)

where $\Omega _u$ is an open set such that $\Omega _u \supset \{u>0\}$ and then applying the method in Propositions 4.4 and 6.9 to the double obstacle problem, $u\in {\tilde{P}}^+_\infty (M)$ to have the classification of global solution.

Theorem A.1

[8] Let u be a global solution of

$$\begin{aligned} Hu =\chi _{\Omega }, \qquad&u \ge 0 \quad \text{ in } {\mathbb {R}}^n \times {\mathbb {R}}, \end{aligned}$$

where $\Omega \supset \{u>0\}$. Then $\partial _t u \le 0$ in ${\mathbb {R}}^n \times {\mathbb {R}}$.

Proposition A.2

Let u be a global solution of

$$\begin{aligned} Hu =\chi _{\Omega _u}, \qquad&u \ge 0 \quad \text{ in } {\mathbb {R}}^n \times {\mathbb {R}}, \end{aligned}$$

where $\Omega _u \supset \{u>0\}$. If we assume that

$$\begin{aligned} \delta _r(u, X)> \epsilon _0 \quad \text { for all } r >0, X\in \partial \Omega _u, \end{aligned}$$

then

$$\begin{aligned} \partial _t u(x,t)\equiv 0 \quad \text { in } {\mathbb {R}}^n \times {\mathbb {R}}\quad \text { and } \quad u=c(x_n^+)^2 \quad \text { in } {\mathbb {R}}^n \times {\mathbb {R}}, \end{aligned}$$

for an appropriate system of coordinates and a positive constant c.

Proof

Suppose that

$$\begin{aligned} \inf _{{\mathbb {R}}^n \times {\mathbb {R}}} \partial _t u=:m<0. \end{aligned}$$

Then, by the same argument as in Proposition 6.7, we have a global solution $u_0$ of

$$\begin{aligned} Hu_0 =\chi _{\Omega _{u_0}}, \qquad&u_0 \ge 0 \quad \text{ in } {\mathbb {R}}^n \times {\mathbb {R}}, \end{aligned}$$

where $\Omega _{u_0}$ is an open set in ${\mathbb {R}}^n \times {\mathbb {R}}$ such that $\{u_0>0\} \subset \Omega _{u_0}$ and $\partial _t u_0 \equiv m$ in $Q_1$ and there is a point $({\tilde{y}}_0, {\tilde{s}}_0)\in \partial Q_1 \cap \partial \Lambda _0$, where $\partial \Lambda _0$ is the boundary of $\Lambda _0$ in ${\mathbb {R}}^n \times {\mathbb {R}}.$ Then,

$$\begin{aligned} u_0(x,t)=m(t-1)+f(x) \quad \text { in } Q_1, \end{aligned}$$

where $f(x):=u_0(x,1)\ge 0$. Moreover, $\partial _t u_0 \equiv m$ in the connected component ${\tilde{\Omega }}(u_0)$ of $Q_1\cup \Omega _{u_0}$ which containing $Q_1$. Since $\partial _t u_0 \equiv m$ in $Q_1$, $f=u_0(x,1) \ge 0$ and $\{u_0>0\} \subset \Omega _{u_0}$, we know that $u_0>0$ in $Q_1$ and $Q_1\cup \Omega _{u_0}=\Omega _{u_0}$.

For $x\in B_1$, suppose that $f(x)>0$. Then, there is a neighborhood $Q_\epsilon (x,1)$ of (x, 1) such that $Q_\epsilon (x,1) \subset {\tilde{\Omega }} (u_0)\subset \Omega _{u_0}$ and

$$\begin{aligned} u_0(x,t)=m(t-1)+f(x) \quad \text { for } t\in (-1,1+\epsilon ). \end{aligned}$$

By applying the argument repeatedly, we know that

$$\begin{aligned} u_0(x,t)>0 \text { and } (x,t) \in \{u_0>0\} \subset {\tilde{\Omega }}(u_0), \quad \text { for all } t \in \left( -1, 1+\frac{f(x)}{-m} \right) \end{aligned}$$

and $\left( x, 1+\frac{f(x)}{-m} \right) \in \partial {\tilde{\Omega }}(u_0)$. Then, the free boundary $\partial {\tilde{\Omega }}(u_0)$ in $B_1\times {\mathbb {R}}$ is represented by $t(x)=1+f(x)/-m$. Since $\nabla u_0=\nabla f(x)$ in $\{(x,t) : t< t(x)\}\cap B_1 \times {\mathbb {R}}$, $\nabla u_0=0$ on $\{(x,t) : t< t(x)\}\cap B_1 \times {\mathbb {R}}$ and $\nabla u_0$ is continuous in ${\mathbb {R}}^n \times {\mathbb {R}}$, we know that $\nabla f(x) =0$ in $B_1$. Hence

$$\begin{aligned} u_0(x,t)=m(t-1)+c_0 \quad \text { on } B_1 \times \left( -\infty ,1+\frac{c_0}{-m} \right) , \end{aligned}$$

for a nonnegative constant $c_0$. by the thickness assumption at $\left( 0, 1+\frac{c_0}{-m} \right)$, we have a contradiction.

Therefore, u is time-independence and by the classification of global solution for elliptic single obstacle problem, u is of the form $c(x_n^+)^2$. $\square$

Proposition A.2 implies that $\partial _t u$ continuously vanishes on the free boundary $\partial \{u=0\}$ for the solution u of the single obstacle problem, (44). By the directional monotonicity, e.g. Sect. 4 of this paper or [8, 12, 14], we have the regularity of the free boundary of the single obstacle problem, (44).

The class (44) is contained in the general classes in [12, 14] and the regularity of the free boundary near the free boundary point 0 of the solutions of (44) was obtained with stronger uniform thickness assumption than (28) in [12, 14], see the introduction of Sect. 6.3.

By Proposition A.2 and the method in Propositions 4.4 and 6.7, we have the classification of global solution in ${\tilde{P}}^+_\infty (M)$.

Proposition A.3

Let $u\in {\tilde{P}}^+_\infty (M)$ and $v:=\psi -u$ and assume that

$$\begin{aligned} \delta _r(u, X)> \epsilon _0 \quad \text { for all } r >0 \text { and } X \in \partial \Omega _u, \end{aligned}$$

$$\begin{aligned} \delta _r(v, X)> \epsilon _0 \quad \text { for all } r >0 \text { and } X \in \partial \Omega _v, \end{aligned}$$

(45)

and

$$\begin{aligned} \delta _r(\psi , X)> \epsilon _0 \quad \text { for all } r >0 \text { and } X \in \partial \Omega _\psi , \end{aligned}$$

(46)

Then,

$$\begin{aligned} \partial _t u(x,t)\equiv 0 \quad \text { in } {\mathbb {R}}^n \times {\mathbb {R}}\quad \text { and } \quad u=c(x_n^+)^2 \quad \text { in } {\mathbb {R}}^n \times {\mathbb {R}}, \end{aligned}$$

for an appropriate system of coordinates and a positive constant c.

The property that $\partial _t u$ continuously vanishes on the free boundary $\partial \{u=0\}$ for the solution $u\in {\tilde{P}}_1^+(M)$ is obtained, by the method in Propositions 4.4 and 6.7. Furthermore, the directional monotonicity for fully nonlinear operator implies the regularity of the free boundary. Since the method and statement of the proof are almost the same as those for $u \in P_1(M)$ and $u \in {\tilde{P}}_1(M)$, we omit the detail.